#8889058

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(3,9)(6,10)(11,15)(14,16)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,10)(11,16)(14,15)(17,19,22,23)(18,20,25,26)(21,29,30,27)(24,31,32,28)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,29,31)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,4,5,13)(2,7,8,12)(3,11,9,15)(6,14,10,16)(17,21,22,30)(18,24,25,32)(19,27,23,29)(20,28,26,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796254);